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This is a question on an active area of research, with lots of work on it (for the general case of algebraically closed fields of char zero). It is historically and conceptually closely connected to K …

Since you mention classification results for $R$-matrices: For finite abelian groups, there is a bijection between the set of universal $R$-matrices of the group hopf algebra $\mathbb C[G]$, the set o …

I do not have access to the article you are citing but I have made a little search and I think the answer to your question is yes (given that we are speaking about a non-compact, connected, semisimple …

Yes there is a complete classification of finite dimensional, simple Lie superalgebras (over $\mathbb{C}$), which -up to a certain extent- goes very much in parallel with the corresponding case of Lie …

Although i have some doubts as to what the OP is exactly looking for (see my comments above), i hope that the following will be of some interest for its purposes. In: $U_q(sl(n))$ Difference Operator …

I am not really sure if this what the OP is looking for but i guess that a closely relevant notion here is that of connected Hopf algebras (i.e HAs which are connected as coalgebras). These are Hopf a …

I will try to provide an answer for a particular case of your last question: Let us consider (following my comment above) the case of $H=\mathbb{CZ}_2$ i.e. the group hopf algebra equipped with its no …

The OP asks more than one different things: the classification of fin dim, semisimple (or not), braided Hopf algebras is still a wide open area (up to my knowledge of course). The classification of …

To any skew-pairing $\lambda:U\otimes H\rightarrow k$, one can associate a hopf algebra $D(U, H)$ (built on $U\otimes H$) which is called the generalized quantum double of $U$ and $H$. (If $H$ is fini …

I think that a general example is the so-called Larson's character, which in a sense ties together the trace and determinant functions. To make the long story short: Let $C$ be a cocommutative bial …

There are some classic results on the classification of the irreducible $D(G)$-modules: If the field is the complex numbers $\mathbb{C}$, it has been shown that a representation of the finite group $G …

If i have correctly understood your question, i think that the answer can be found at M. D. Gould, R. B. Zhang, Classification of all star and grade star irreps of gl(n|1), J. of Math. Phys., 31, 15 …

If the question is set on the level of mentioning important "theorems" or "computations" or "results" which wouldn‘t have been proved without the development of string theory i think one could …

Regarding the "what is happening in the super case"; yes i agree that in some sense, it has to do with the odd simple roots but i think it is deeper than that: In the case of semisimple, complex, Lie …

This is a very interesting question. I have also made some search but i have not found this result explicitly mentioned somewhere in the literature. However, i remember i have heard such a claim in th …